Irrational Numbers Class 9 Maths Chapter 1 Number System

Number Systems: Rational and Irrational Numbers on the Real Number Line

The number system is the foundation of all mathematics. This chapter in CBSE Class 9 Mathematics extends the students' understanding of numbers from the familiar rational numbers to the broader category of irrational numbers, and together they form the complete set of real numbers. Students learn to represent numbers on the number line through successive magnification, understand the density of rational and irrational numbers, and perform operations with real numbers in decimal form.

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Rational numbers have decimal expansions that either terminate (like 0.75 = 3/4) or repeat periodically (like 0.333... = 1/3). An irrational number is a real number that cannot be expressed as p/q — its decimal expansion is non-terminating and non-recurring. Common examples include √2 ≈ 1.41421356..., √3 ≈ 1.73205080..., π ≈ 3.14159265..., and the golden ratio φ ≈ 1.61803398... The irrationality of √2 is one of the most famous proofs in mathematics. It is proved by contradiction: assume √2 is rational, so √2 = p/q where p and q have no common factors. Then 2 = p²/q², so p² = 2q², meaning p is even. Let p = 2k, then 4k² = 2q², so q² = 2k², meaning q is also even. This contradicts the assumption that p and q have no common factors, so √2 cannot be rational. Every real number has a unique decimal expansion on the number line. To locate irrational numbers like √2, we use the property that a line segment of length √2 can be constructed using the Pythagorean theorem: a right-angled triangle with both legs of length 1 has a hypotenuse of length √2. We can then mark this length on the number line starting from the origin.

Any real number can be represented on the number line through the process of successive magnification. For example, to locate 2.3̄3̄ = 2.333..., we first identify that it lies between 2 and 3, then between 2.3 and 2.4, then between 2.33 and 2.34, and so on — each step zooms in on a smaller interval that contains the number. The operations on real numbers follow the same rules as for rational numbers. The sum, difference, product, and quotient (with a non-zero divisor) of two real numbers is always a real number. However, if we try to add, subtract, multiply, or divide two irrational numbers, the result may be rational or irrational. For example, √2 × √3 = √6 is irrational, but √2 × √8 = √16 = 4 is rational. Similarly, (3 + √2) + (3 − √2) = 6 is rational, showing that the sum of two irrational numbers can be rational. Rationalising the denominator is the process of eliminating irrational numbers from the denominator of a fraction. For example, 1/√2 = √2/2 (multiply numerator and denominator by √2), and 1/(√3 − √2) = (√3 + √2)/((√3)² − (√2)²) = (√3 + √2)/1 = √3 + √2 (multiply by the conjugate). The laws of exponents with rational exponents extend to real numbers: a^(m/n) = ⁿ√(aᵐ). These properties are essential for simplifying expressions involving roots and for solving equations involving irrational quantities.

• Rational numbers can be expressed as p/q (terminating or repeating decimals); irrational numbers cannot (non-terminating, non-recurring decimals).
• The irrationality of √2 is proved by contradiction: assuming √2 = p/q with no common factors leads to p and q both being even.
• Every real number (rational or irrational) has a unique position on the number line; irrational numbers are located using geometric construction or successive magnification.
• The sum, difference, product, or quotient of two irrational numbers can be either rational or irrational depending on the numbers involved.
• Rationalise the denominator by multiplying by the conjugate: 1/(√a − √b) = (√a + √b)/(a − b).